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Selecting the Runoff Pair Selecting the runoff pair James Green-Armytage New Jersey State Treasury Trenton, NJ 08625 [email protected] T. Nicolaus Tideman Department of Economics, Virginia Polytechnic Institute and State University Blacksburg, VA 24061 [email protected] This version: May 9, 2019 Accepted for publication in Public Choice on May 27, 2019 Abstract: Although two-round voting procedures are common, the theoretical voting literature rarely discusses any such rules beyond the traditional plurality runoff rule. Therefore, using four criteria in conjunction with two data-generating processes, we define and evaluate nine “runoff pair selection rules” that comprise two rounds of voting, two candidates in the second round, and a single final winner. The four criteria are: social utility from the expected runoff winner (UEW), social utility from the expected runoff loser (UEL), representativeness of the runoff pair (Rep), and resistance to strategy (RS). We examine three rules from each of three categories: plurality rules, utilitarian rules and Condorcet rules. We find that the utilitarian rules provide relatively high UEW and UEL, but low Rep and RS. Conversely, the plurality rules provide high Rep and RS, but low UEW and UEL. Finally, the Condorcet rules provide high UEW, high RS, and a combination of UEL and Rep that depends which Condorcet rule is used. JEL Classification Codes: D71, D72 Keywords: Runoff election, two-round system, Condorcet, Hare, Borda, approval voting, single transferable vote, CPO-STV. We are grateful to those who commented on an earlier draft of this paper at the 2018 Public Choice Society conference. 2 1. Introduction Voting theory is concerned primarily with evaluating rules for choosing a single winner, based on a single round of voting. Within that framework, a myriad of alternatives to plurality (or “first past the post”) have been discussed over the years, such as Borda, Hare, Dodgson, Nanson, Baldwin, Black, Kemeny, Coombs, minimax, approval voting, ranked pairs, anti-plurality, beatpath and Condorcet-Hare, as well as many others.1 However, many elections to identify a single winner are conducted using a two-round, or runoff system. For example, of the 113 countries that hold presidential elections, 87 conduct their elections using a runoff system.2 While it is true that something resembling a runoff system can be implemented with a single round of voting using ballots on which voters have ranked the candidates, that is not fully equivalent to a runoff system because it does not give citizens the option of changing their reported preferences, or their decisions about whether or not to vote, between rounds. Thus, one function of a runoff is to concentrate the attention of citizens on the relative merits of the two finalists. When citizens know who the finalists are, they have stronger motivations to acquire information about them, ensuring that the eventual winner will undergo close scrutiny during the campaign and will have a majority against the one remaining alternative. We argue that a traditional runoff system is a member of a family of two-round voting rules that is distinct from the family of one-round systems, and that the family of two-round systems invites a distinct analysis that has not been undertaken previously. This paper begins 1 See Borda (1784), Hare (1865), Dodgson (1876), Nanson (1882), Baldwin (1926), Black (1958), Kemeny (1959), Coombs (1964), Simpson (1969), Brams and Fishburn (1978), Tideman (1987), Saari (1990), Schulze (2003) and Green-Armytage et al. (2016), respectively. Levin and Nalebuff (1995) and Tideman (2006) provide surveys. Note that what we call Hare is the single-winner version of single transferable vote, which also is known as the alternative vote, instant runoff voting and ranked choice voting. Dodgson (1876) proposed more than one election procedure, but here we mean the well-known one referred to as the “Dodgson rule” by Levin and Nalebuff (1995, pp. 24–25) and by Tideman (2006, pp. 196–199). Kemeny also is known as Kemeny-Young, because of Young (1988). Minimax also is known as Simpson, Simpson-Kramer, successive reversal, min-max, and maximin. Beatpath also is known as clone- proof Schwartz sequential dropping, or the Schulze method. 2 International Institute for Democracy and Electoral Assistance (2018). 3 that analysis. We define a “runoff pair selection rule” as follows: In the first round, voters submit ballots, which could be plurality, ranked, approval or range ballots, and those ballots are used to choose two and only two candidates, who will be considered in the second round. In the second round, voters choose between the two remaining candidates by a simple majority vote.3 Our goal is to evaluate such rules relative to one another. That is, given the premise that two candidates must be chosen for a runoff, we ask: What is the best method for choosing them? To confront that question, we must first recognize that what is ‘best’ in the context of choosing a runoff pair might be different from what is ‘best’ in the context of choosing a single winner. For example, a society may wish to choose the two candidates who would, individually, give voters the most utility if elected. Alternatively, a society may wish to choose the two candidates who are most broadly representative of the electorate as a whole. When a Condorcet winner exists, the society may wish to ensure that that candidate is included in the runoff pair. The preferred rule for selecting a runoff pair may vary depending on what goals society has. In this paper, we evaluate nine possible rules for selecting a runoff pair on the basis of four criteria. Neither the rules nor the criteria are intended to be exhaustive; rather, they are intended to provide an initial analysis and stimulate readers’ imaginations as to what other analyses might be worthwhile. The remainder of the paper is organized as follows: Section 2 defines the rules to be evaluated, Section 3 defines the criteria used to evaluate them, Section 4 describes the data used to perform our evaluations, Section 5 presents the results; Section 6 concludes. The Appendix defines additional rules and subjects them to preliminary evaluation. The Supplemental 3 Many runoff systems declare a winner in the first round if a single candidate receives a majority of the vote, thus canceling the second round. In the interest of simplicity, we do not treat such systems separately. In addition, some two-round systems (such as the system used to elect France’s Assemblée nationale) sometimes field more than two candidates in the second round. To avoid complications, we do not consider such a possibility. 4 Appendix, which is available online, proposes additional criteria, presents results from the spatial model with different parameters, and compares some of the runoff procedures with their single-round counterparts in terms of resistance to strategy. 2. Runoff pair selection rules 2.1. Plurality rules 2.1.1. Plurality. This is the traditional rule. The runoff pair consists of the candidates with the top-two plurality scores. 2.1.2. Hare. The Hare rule eliminates candidates one by one on the basis of which candidate has the fewest first-place votes; each time a candidate is eliminated, each vote for that candidate is reassigned to the not-yet-eliminated candidate ranked next on that ballot. Application of the Hare rule chooses as the runoff pair the last two candidates who remain after the others have been eliminated. 2.1.3. Single transferable vote (STV). When used as a runoff pair selection rule, the single transferable vote (STV) rule4 is like the Hare (last two) rule except that when a candidate receives more than a “quota” of votes, a fraction of the votes received are sent to the next candidates in those voters’ rankings, so that in the end no candidate has more than a quota of votes. Because two candidates are to be chosen, a quota is one-third of the votes.5 2.2. Utilitarian rules 2.2.1. Borda. Each candidate’s Borda score is the sum of points earned, assigned in descending 4 Many versions of STV have been proposed; for a review, see Tideman (1995). The version we use performs fractional transfers of ballots when the first candidate achieves a quota and then divides a voter’s ballot strength evenly among candidates when the voter lists them as being tied as the most preferred among non-elected, non-eliminated candidates. 5 Thus, we use a modified “Droop quota”: specifically, we use the number of voters divided by three, plus 1/1,000,000. Droop’s definition of the quota named for him was 1 + the integer part of [votes ÷ (positions + 1)]. 5 order from N – 1, where N is the number of candidates on the ballot; each position on a voter’s ballot is worth the number of positions below that position. The rule selects the two candidates with the highest Borda scores. 2.2.2. Approval. Each voter gives each candidate either one point or zero points. The runoff pair consists of the two candidates with the highest point totals. In our simulations, we assume that a voter approves a candidate if and only if that candidate gives the voter utility greater than or equal to their average utility for all candidates. 2.2.3. Range. Each voter gives each candidate a score on a closed interval. The runoff pair consists of the two candidates with the greatest sums of scores. In our simulations, scores are utilities, bounded between zero and one. 2.3. Condorcet rules 2.3.1. Condorcet-Hare. For electing one candidate, the Condorcet-Hare rule is that if a Condorcet winner exists, then that candidate is selected.
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